Races and games

Race :-

When a contestant participate in any contest , then a contest of speed in running, driving, riding sailing or rowing over a specified distance is called race.

Race Course :-

The ground or path on which contests are organised in a systematic way , is called a race course.

Starting Point :-

The exact point or place from where a race begins is called the starting point.
For example :- If A and B are the contestants for a 100 m race and A has to cover 100 m , while B has to cover ( 100 - 20 ) = 80 m

Winning Post (or Goal) :-

The point where the race finishes is called the winning post or finishing point or goal.
For example :- In a 800 m race , A gives some start to B and this makes the length of race for B 725 m .What start does B get from A ?
Sol :- Start given by A to B = ( 800 - 725 )m = 75 m

Dead-heat Race :-

If all the persons contesting a race reach the finishing point exactly at the same time, then the race is called a dead-heat race.

Winner :-

The person who first reaches the finishing point is the winner.
For example :- If in game of 100 points means that the contestant who scores 100 points first , is declared the winner .While who scores 85 points , is second .

Some Facts about Race :-

1. Let two contestants are A and B .If A beats B by d m , then

Distance covered by A who is winner = L m
Distance covered by B who is loser = ( L - d ) m

Example 1. In a race of 100 m , A gives B a start of 25 m .What distance will be covered by B ?
Sol :- Here , Distance covered by A = 100 m and d = 25 m
Using the above formula , we have
Distance covered by B = 100 - 25 = 75 m

2. Two contestants are A and B .If B begins from d m ahead of A , then
A starts at the starting point , B starts d m ahead from the starting point at the same time .

Distance covered by B = ( L - d ) m

Example 2. In a race of 200 m , If B begins from 50 m ahead of A at same time .What distance will be covered by B ?
Sol :- Here , L = 200 m and d = 50 m
Using the above formula , we have
Distance covered by B = 200 - 50 = 150 m

3. A and B are two contestants .If A beats B by T seconds , then
A and B both starts from starting point .

Time taken by A who is winner = Time taken by B who is loser - T

Example 3. In 1 km race , A beats B by 18 m or 9 s .Find the A' s time over the course ?
Sol :- Given , B covers 18 m or 9 s .
B' s time over the course = ( 9 / 8 ) x 1000 = 500 s
Using the above formula ,
A's time over the course = 500 - 9 = 491 s

4. Let A and B are two contestants .A game in which the participant scoring 100 points first is the winner .

5. If both of the contestants A and B get at finishing point at the same time , then

Difference in the time of defeat = 0
, Difference in distance of defeat = 0

Some Useful Shortcut Method :-

1. If A is n times as fast as B and A gives B a start of d metres .If both A and B reach the winning post at the same time, then the length of the race course must be

d n metres

( n - 1 )

Example 4. A is 3 times as fast as B. If A gives B a start of 80 m. How long should the race course be so that both of them reach at the same time?
Sol :- Here , n = 3 times and d = 80 m
Using the above given formula , we have

Length of the race course = 80 x	3	meters
	( 3 - 1 )

= 80 x	3	= 40 x 3 = 120 metres.
	2

Hence , length of the race course is 120 metres.

2. If A can run d m race in T₁ seconds and B in T₂ seconds ,then A beats B by a distance is

d x ( T₂ - T₁ ) meters

T₂

where T₁ < T₂,
Example 5. A can run 200 m in 20 seconds and B in 25 seconds. By what distance A beats B?
Sol :- Here , T₁ = 20 seconds , T₂ = 25 seconds and d = 200 m
Using the above given formula ,

A beats b by a distance = 80 x	200	x ( 25 - 20 ) = 8 x 5 = 40 metres
	25

3. Let A and B are two contestants.If A is winner and B is loser , then the following relation will be

Time taken by A = Time taken by B

Distance covered by B Distance covered by A

⇒ Time taken by A = ( Difference of winning time + initial time )

Distance covered by B ( Difference of winning distance + initial distance )

Example 6. In a 1000 m race , P beats Q by 72 metres in 12 seconds .Find the P's time over the course .
Sol :- Here , L = 1000 m and d = 72 metres
Difference of winning time = 12 seconds
Difference of winning distance = 72 metres
Using the above given formula , we get

	Time taken by A	=	12
	( 1000 - 72 )		72

	Time taken by A	=	12
	928		72

Time taken by A = 928 ×	12	=	464
	72		3

= 154	2	meters
	3

Hence , Time taken by A is 154² /₃ metres.

4. If three contestants A , B and C are participants in a race of L m , A contestant beats B contestant and C contestant by distances of d₁₂ and d₁₃ metres respectively and B contestant beats the C contestant by a distance of d₂₃ , then we get the following relation between them

( L - d₁₂ )d₂₃ = L ( d₁₃ - d₁₂ )

Example 7. A , B and C are three contestants in a 500 m race .If A can give B a start of 20 m and A can give C a start of 32 m. How many metres start can B give to C ?
Sol :- Here , d₁₂ = 20 m , d₁₃ = 32 m , d₂₃ = ?
and L = 500 m
Using the above given formula ,
( 500 - 20 )d₂₃ = 500 ( 32 - 20 ) = ( 25 - 20 )
⇒ 480d₂₃ = 500 x 12

d₂₃ =	( 500 x 12 )
	480

d₂₃ =	50	= 12.5 metres
	4

5. A , B and C are any three contestants .If in a race of L₁ m , A contestant beats B contestant by a distance of d₁₂ , in a race of L₂ m B contestant beats C contestant by a distance of d₂₃ and in a race of L₃ m , A contestant beats C contestant by a distance of d₁₃ , then for a race of L m

A₁₂ = d₁₂ × L , A₂₃ = d₂₃ × L and A₁₃ = d₁₃ × L

L₁ L₂ L₃

Now , we get the following relation

( L - A₁₂ )A₂₃ = L ( A₁₃ - A₁₂ )

Example 8. In a race of 1000 m , A can beat by 80 m and in a race of 640 m , B can beat C by 160 m .By how many metres will A beat C in a race of 900 m ?
Sol :- Here , L₁ = 1000 m , d₁₂ = 80 m , d₂₃ = 160 m , L₁ = 640 m , L = 900 m and A₁₃ = ?
Using the above given formula ,

A₁₂ =	80	× 900 = 8 x 9 = 72 m
	1000

A₂₃ =	160	× 900 =	900	= 250 m
	640		4

and A₁₃ = ?
Now , ( 900 - 72 ) x 250 = 900 ( A₁₃ - 72 )
⇒ 828 x 250 = 900A₁₃ - 900 x 72 ⇒ 207000 = 900A₁₃ - 64800
⇒ 900A₁₃ = 207000 + 64800

⇒ A₁₃ =	271800	= 302 m
	900

Hence , A will beat C by a distance of 302 m .

Example 9. Arun and Bhaskar start from place P at 6 : 00 am and 7 : 30 am, respectively and run in the same direction. Arun and Bhaskar run at 8 km/h and 12 km/h. respectively. Bhaskar overtakes
Arun at ?

Sol :- Distance between Arun and Bhaskar at 7 : 30 am = 8 x	11	= 12 km
	2

Time taken by Bhaskar in covering a distance of 12 km =	12	= 3 h
	( 12 - 8 )

∴ Required time = 10 : 30 am
Example 10. A 10 km race is organised at 800 m circular race course. P and Q are the contestants of the race. If the ratio of the speed of P and Q is 5 : 4, how many times will the winner overtake the loser ?
Sol :- Speed of P : Speed of Q = 5 : 4
Time taken by P to cover 5 rounds = Time taken by Q to cover 4 rounds

Distance covered by P in 5 rounds = 5 x	800	= 4 km
	1000

Distance covered by Q in 4 rounds = 4 x	800	=	16	Km
	1000		5

In 5 rounds, P will overtake Q every time. It means that after covering 4 km, P will overtake Q one time .

∴ After covering 10 km P will overtake Q =	1	x 10 =	21	times
	4		2

Example 11. In a race of 600 m, A can beat B by 60 m and in a race of 500 m, B can beat C by 50 m. By how many meters will A beat C in a race of 400 m?
Sol :- Clearly, if A runs 600 meters, B runs = 540 m.

∴ If A runs 400 m, B runs = 540 ×	400	= 540 ×	4	= 90 × 4 = 360 m
	600		6

Again, when B runs 500 m, C runs = 450 m

∴ When B runs 360 m, C runs = 450 ×	360	= 45 ×	36	= 9 × 36 = 324 m
	500		5

∴ A beats C by 400 - 324 = 76 m.

Time taken by A	=	Time taken by B
Distance covered by B	=	Distance covered by A

⇒	Time taken by A	=	( Difference of winning time + initial time )
	Distance covered by B		( Difference of winning distance + initial distance )

A₁₂ =	d₁₂	× L , A₂₃ =	d₂₃	× L and A₁₃ =	d₁₃	× L
	L₁		L₂		L₃